Write $\frac{1+\sqrt{-32}}{4}$ as a complex number in the form $a+b i$, where $a$ and $b$ are real numbers.
Answer (1)
Rewrite − 32 as $4
\sqrt{2}i$.
Substitute into the original expression: 4 1 + 4 2 i .
Separate the real and imaginary parts: 4 1 + 4 4 2 i .
Simplify to obtain the final answer: 4 1 + 2 i .
Explanation
Understanding the Problem We are given the complex number 4 1 + − 32 and we want to express it in the form a + bi , where a and b are real numbers.
Rewriting the Square Root First, we rewrite − 32 using the imaginary unit i , where i = − 1 . We have − 32 = 32 ⋅ ( − 1 ) = 32 ⋅ − 1 = 32 i .
Simplifying the Radical Next, we simplify 32 . Since 32 = 16 ⋅ 2 , we have 32 = 16 ⋅ 2 = 16 ⋅ 2 = 4 2 .
Substituting Back Now we substitute this back into our expression: 4 1 + − 32 = 4 1 + 4 2 i .
Separating Real and Imaginary Parts We separate the real and imaginary parts: 4 1 + 4 2 i = 4 1 + 4 4 2 i .
Simplifying the Expression Finally, we simplify the expression: 4 1 + 4 4 2 i = 4 1 + 2 i . Thus, the complex number in the form a + bi is 4 1 + 2 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltage and current can be represented as complex numbers, making calculations much easier. For example, you can use complex impedance to find the total impedance in a circuit with resistors, capacitors, and inductors. This helps engineers predict how the circuit will behave and optimize its performance.
